46 problems

Let $\Theta^\infty=\Theta^{L(\Gamma^\infty,\mathbb{R})}$ and let $L(\mathcal{A}^\infty)$ be the model generated by $\mathcal{A}^\infty$. Square-sequence conjecture. … The claim is…

Assume ${\sf{AD}}^+$. Let $\kappa$ be a regular Suslin cardinal, and let ${\mathbf{\Gamma}}$ be the pointclass of $\kappa$-Suslin sets. Suslin-cardinal unreachable successor conjec…

Let $A_{(\alpha,n)}$ be the sets defined from $J_{\alpha}(\mathbb{R})$, and let a mouse set mean a set $A\subseteq\mathbb{R}$ of the form $A=\mathbb{R}\cap\mathcal{M}$ for some cou…

Let $({\mathcal{P}},\Psi)$ be a hod pair such that … Let $\kappa$ be the least strong cardinal which reflects the class of strong cardinals, and let…

Let $C(\mathop{\text{aa}})$ and $C^2(\omega)$ denote the inner models defined in the paper, with $C(\mathop{\text{aa}})$ the model generated using the relevant class of structures…

Let $C^2(\alpha9\omega)$ denote the inner model obtained from second-order definability over $\alpha9\omega$, let $\text{CH}$ denote the continuum hypothesis, and let $M_n$ denote th…

Solovay-sequence conjecture. Then $N\vDash\mathsf{ZF}$, $\Theta^N=\theta_\alpha$, $N$ satisfies that every cardinal $\eta\geq\Theta$ has regular successor $\eta^+$, and, for every…

Cardinal-regularity conjecture. Then

Let $\mathrm{EEA}$ denote the Extender Embedding Axiom, and let $V=\mathrm{Ultimate-}L$ denote the corresponding Ultimate-$L$ axiom. Woodin's Ultimate-$L$ conjecture. The theory ……

Solovay-sequence conjecture. Under these assumptions, $V_\zeta^{{\sf{HOD}}_X}$ is a model of

Least-cardinal structural reflection conjecture. A cardinal $\kappa$ is the least cardinal satisfying some large-cardinal notion if and only if $\kappa$ is the least cardinal satisfy…

Assume $\mathrm{ZF}+\mathsf{AD}^{L(\mathbb R)}$. Let $M$ be a $(0,\omega_1)$-iterable countable $\omega$-small premouse satisfying “$\omega_1$ exists”. Then there are $\gamma$,…

Let $(\mathcal{V},\Omega)$ be an excellent least branch hod pair with $\mathcal{V}\models\mathsf{ZFC}$, let $\delta$ be a Woodin limit of Woodin cardinals in $\mathcal{V}$, and let…

Let $V$ satisfy $obreaktextsf{AD}{\mathbb R}+ obreaktextsf{NLE}+V=L({wp}(\mathbb R))$. Let $obreakDeltasubseteq{wp}(\mathbb R)$ and let…

Let $obreaktextsf{AD}{\mathbb R}$ denote determinacy for real games. Suppose that for every set $Bsubseteqmathbb R$ there is a function $F$ as in Theorem n*x such that $F$ is not p…

Let $obreaktextsf{AD}^+$ denote the strengthened axiom of determinacy. For a function $F$ as in the paper's Theorem n*x, call $F$ pe-bad when, for a cone of reals $x$, the sup…

Let $obreaktextsf{AD}^+$ denote the strengthened axiom of determinacy, and let $obreaktextsf{NLE}$ denote the no-long-extender assertion. Let $obreaktextsf{HPC}$ denote hod-pair…

Let $obreaktextsf{AD}^+$ denote the strengthened axiom of determinacy, and let $obreaktextsf{NLE}$ be the no-long-extender assertion. Let $obreaktextsf{LEC}$ denote $L[vec E]$-c…

Covering with Chang Models conjecture. The asserted transitive model $M$ exists with all the listed properties. This conjecture is intended to overcome limitations of the core mode…

Assume $ZF + AD + DC + V=L(\mathbb{R})$. Let $\alpha\in ON$ and $n\in\omega$. For each $x\in\mathbb{R}$, define $\Gamma_x$ to be the collection of all pointsets $A$ such that for s…

Assume $ZF + AD + DC + V=L(\mathbb{R})$. A mouse set is a set $X\subseteq\mathbb{R}$ for which there is an $\omega_1+1$-iterable premouse $M$ with $X=M\cap\mathbb{R}$. Let $\Gamma$…

Let $LSA$ abbreviate the assertion that $AD^+$ holds and there is a largest Suslin cardinal belonging to the Solovay sequence. Let $\vec{E}$ be an extender sequence and write…

Assume $AD^+$ and that there is a largest Suslin cardinal which is a member of the Solovay sequence. A transitive model $M$ of $ZF$ is a minimal model of $\Theta strong$ when…

Covering conjecture. For every such $\lambda$, $A$, and $B$, there is an $A$-iteration set $X\in{\wp}_\kappa(\lambda)$ such that $B\subseteq X$. This asserts a covering property fo…

Assume ${\sf{NLE}}$ and suppose there are unboundedly many Woodin cardinals and strong cardinals. Let $\kappa$ be a limit of Woodin cardinals and strong cardinals such that either…